Problem: The sum of two angles is $78^\circ$. Angle 2 is $90^\circ$ smaller than $3$ times angle 1. What are the measures of the two angles in degrees?
Answer: Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 78}$ ${y = 3x-90}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${3x-90}$ for $y$ in the first equation. ${x + }{(3x-90)}{= 78}$ Simplify and solve for $x$ $ x+3x - 90 = 78 $ $ 4x-90 = 78 $ $ 4x = 168 $ $ x = \dfrac{168}{4} $ ${x = 42}$ Now that you know ${x = 42}$ , plug it back into $ {y = 3x-90}$ to find $y$ ${y = 3}{(42)}{ - 90}$ $y = 126 - 90$ ${y = 36}$ You can also plug ${x = 42}$ into $ {x+y = 78}$ and get the same answer for $y$ ${(42)}{ + y = 78}$ ${y = 36}$ The measure of angle 1 is $42^\circ$ and the measure of angle 2 is $36^\circ$.